A rectangular auditorium seats 2100 people. The number of seats in each row exceeds the number of rows by 8. Find the number of seats in each row.

If D be a region bounded by a simple closed path C in the xy-plane, then the centroid of the quarter circle is  (4a/3π , 4a/3π)   .

In the question ,

it is given that D is the region bounded by a simple closed path C in the xy-plane .

the radius of the circle is = "a" ,

let the equation of the circle be x² + y² = a² ,

So , the area of the quarter circle is (A)  = (1/4)*πa²

The coordinate of the centroid x will be :

x = [tex]\frac{1}{2A} \int\limits x^{2} dy[/tex]

x = [tex]\frac{1}{2(\frac{\pi a^{2} }{4})} \int\limits^a_0 {a^{2} -y^{2} } \, dy[/tex]

Simplifying further ,

we get ,

x = 2/πa² [{a²(a) - a³/3} - 0 ]

x = 2/πa² [a³ - a³/3 ]

x = 4a/3π

The coordinate of the centroid y will be :

y = [tex]\frac{1}{2A} \int\limits y^{2} dx[/tex]

y = [tex]\frac{1}{2(\frac{\pi a^{2} }{4})} \int\limits^a_0 {a^{2} -x^{2} } \, dx[/tex]

Simplifying further ,

we get ,

y = 2/πa² [{a²(a) - a³/3} - 0 ]

y = 2/πa² [a³ - a³/3 ]

y = 4a/3π

Therefore , the coordinates of the centroid is (4a/3π , 4a/3π) .

The given question is incomplete , the complete question is

Let D be a region bounded by a simple closed path C in the xy-plane. The coordinates of the centroid x, y of D are x = [tex]\frac{1}{2A} \int\limits x^{2} dy[/tex]  ,  y = [tex]\frac{1}{2A} \int\limits y^{2} dx[/tex] ?  

Find the centroid of a quarter-circular region of radius a.

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Answer:

JN

Step-by-step explanation:

On solving the question, by the help of angle sum property of triangle, we got that x= 14

Having three edges and three vertices, a triangle has three sides. The most significant characteristic of triangles is the fact that their interior angles add up to 180 degrees.

Scalene, isosceles, and equilateral triangles are the three types of triangles that can have three sides.

Now, value of x is -

(x+27)° + 70° + (x + 55)° = 180° ( Sum of  al angles in a triangle is 180°)

2x + 152 = 180°

2x = 28

x=14

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The length of the hypotenuse AC is √(h² + 4).

Pythagorean theorem states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

We have,

Right triangle ACD.

AC² = CD² + AD²

AC² = h² + 2²

AC² = h² + 4

AC = √(h² + 4)

Thus,

AC is √(h² + 4).

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Answer: C: $7,000.00

Step-by-step explanation:

The credit limit which gives an acceptable debt ratio is $7000.

A credit limit is the most money you are permitted to spend on a credit card or line of credit by a lender. Knowing your limit, however, does not imply that going for it is a wise decision.

Given:

Owed Amount = $ 3450.0

So, Debt Ratio= (Total liabilities) ÷ (Total amount of worth possessed)

A debt ratio of approximately 0.5 is considered to be reasonable.

Money Owed by you = $ 3450.00

0.5 = 3450/credit limit

credit limit = 3450 × 10/5

credit limit = 6900

So, the Credit limit which gives Acceptable Debt of $ 3450.00 when debt ratio is approximately more or less than 0.5 is $7000.

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The equation that represents the the total amount he paid, y, for x pounds of potatoes is y = 0.64x.

Dontrelle bought [tex]5\frac{1}{2}[/tex] pounds of potatoes for $3.52.

The equation that can be used to represent the the total amount he paid, y, for x pounds of potatoes is as follows:

rate = amount paid / weight in pounds

amount paid = $3.52

weight of potatoes = 11 / 2 pounds

rate = 3.52 ÷11 / 2

rate = 3.52 × 2  / 11

rate = 7.04 / 11

rate = 0.64 dollars per pound.

Therefore, the equation is y = 0.64x

where

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The equation that is quadratic in all the equations is 6(2x+4)²=(2x+4)+2

A quadratic equations defined as any equation that can be rearranged in standard form as ax²+bx+c=0 or -b±[tex]\frac{\sqrt{} b^{2}-4ac }{2a}[/tex]

From the list of equations the one that carries the standard form for a quadratic equation is

6(2x+4)²=(2x+4+)+2

Expand the brackets to have

6[(2x+4)(2x+4)=(2x+4)+2

6[4x²+8x+8x+16]=(2x+4)+2

Collecting like terms and opening the brackets further;

6[4x²+16x+16]=2x+4+2

=24x²+96x+96=2x+6

Collecting like terms we have

24x²+96x-2x+96-6=0

⇒24x²94x+90=0

In conclusion, the equation carries 2 as its highest power therefore, it is a quadratic equation.

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When renting a car for $40.49 with a fixed mileage charge of $0.05, a businessperson can go 1590.2 miles.

An expression, often known as a mathematical expression, is a finite collection of symbols that are well-formed in accordance with context-dependent principles. Mathematical expressions consist of at least two numbers or variables, at least one arithmetic operation, and a statement. It's possible to multiply, divide, add, or subtract with this mathematical operation. An expression in mathematics is made up of numbers, variables, and functions (such as addition, subtraction, multiplication or division etc.) You can think of expressions as being comparable to phrases.

Here,

The cost of car rent=$40.49

Cost for per mile=$0.05

Let him travel x miles,

40.49+0.05x=120

0.05x=120-40.49

0.05x=79.51

x=79.51/0.05

x=1590.2 miles

The business person can travel 1590.2 miles when the rent of car cost $40.49 fixed and $0.05 for every mile travelled.

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Answer:

  9

Step-by-step explanation:

You want to know the least number of visits Jonathan must make to a movie theater to have at least 135 points if he has 25 points and earns 13.5 for each visit.

After x visits to the theater, Jonathan will have earned 13.5x points in addition to the signing bonus. His total will be ...

  13.5x +25

Jonathan wants that total to be at least 135:

  13.5x +25 ≥ 135

  13.5x ≥ 110 . . . . . . . subtract 25

  x ≥ 110/13.5 ≈ 8.148

The smallest integer greater than 8.148 is 9.

Jonathan needs to make at least 9 visits to earn a free movie ticket.

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A hypothesis is an assumption, an idea that is proposed for the sake of argument so that it can be tested to see if it might be true.

According to the statement, the value of Z-Stat is +2.10. And we must find the P value when H0 is rejected only in the upper tail. The Z-stat is defined as the Z-score, which indicates how much a given value differs from the standard deviation.

The Z-score, also known as the standard score, is the number of standard deviations a given data point is above or below the mean.

And we know that the P-Value is calculated by converting your statistic into a Z-Score.

As a result, the p value is 0.0179.

So, When ZSTAT = +2.10, the resulting P-Value is 0.0179.

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Total profit - Total cost = Profit

The profit made is $1078.

Option C is the correct answer.

An equation is a mathematical statement that is made up of two expressions connected by an equal sign.

Example:

2x + 7 = 8 is an equation.

We have,

The total profit, in dollars per day, after x days can be modeled by the function

R(x) = 3x³ – 21x² + 21x + 45.

The total cost, in dollars per day, after x days, can be modeled by the function

C(x) = x² – 2x – 3.

Now,

(R - C) (10):

= R(10) - C(10)

= 3x³ – 21x² + 21x + 45 - x² + 2x + 3

= 3x³ - 22X² + 23x + 48

= 3 x 1000 - 22 x 100 + 230 + 48

= 3000 - 2200 + 230 + 48

= $1078

Thus,

Total profit - Total cost = Profit

The profit made is $1078.

Option C is the correct answer.

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The indicated side of the right angle triangle is 183.

A right triangle is a triangle that has one of it's angles as 90 degrees. A right triangle sides are named according to the position of the angles.

The sides name are as follows:

let's find the value of side h using the trigonometric ratios as follows

tan ∅ = opposite / adjacent

Therefore,

tan 18.8  = h / x + 423

cross multiply

(x + 423) tan 18.8 = h

h = x tan 18.8 + 423 tan 18.8

h = 0.34042776856x + 144.000946101

tan 57.8 = h / x

cross multiply

x tan 57.8 = h

h = 1.58797305139x

Therefore,

1.58797305139x = 0.34042776856x + 144.000946101

1.58797305139x - 0.34042776856x = 144.000946101

1.24755305139x = 144.000946101

x = 144.000946101 / 1.24755305139

x = 115.426674549

Therefore,

h =  1.58797305139(115.426674549)

h = 183.294448434

h = 183

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Answer:  

~ Sarah = b

!!!Michael = 1/2*b + 3

~~David = b/2 + 8

!!!Hannah = 3b + 28

~~~~~Sarah = 3b/2 + 14

By solving the given equation, we know that Courtney withdrawals $25.75 each time from her bank account.

A mathematical statement known as an equation is made up of two expressions joined together by the equal sign.

A formula would be 3x - 5 = 16, for instance.

When this equation is solved, we discover that the value of the variable x is 7.

So, we have the equation:

400-3x=322.75

Now, solve the equation for x to get the amount Courtney withdrawals each time:

400-3x=322.75

-3x = 322.75 - 400

-3x = -77.25

x = -77.25/-3

x = $25.75

Therefore, by solving the given equation, we know that Courtney withdrawals $25.75 each time from her bank account.

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Complete question:
Courtney made 3 withdrawals of equal amounts from her savings account. The equation below can be used to determine x, the amount in dollars of each withdrawal.

400-3x=322.75

What was the amount in dollars of each of Courtney's withdrawals?

answer choices

a. $24.09

b. $25.75

c. $74.25

d. $77.25

We take the inverses of the function and get the results as follows:

[tex]f(1)=1+1=2\\f^{-1}(1)=\sqrt{1-1}=0\\f^{-1}(f(1))=\frac{\sqrt{\sqrt{4(1)+1} -3}}{\sqrt{2} } \\f(f^{-1} (1))=1[/tex],  [tex]f(2)=4+1=5\\f^{-1}(2)=\sqrt{2-1}=1\\f^{-1}(f(2))=\frac{\sqrt{\sqrt{4(2)+1} -3}}{\sqrt{2} } \\f(f^{-1} (2))=2[/tex], [tex]f(3)=9+1=\sqrt{10} \\f^{-1}(3)=\sqrt{3-1}=\sqrt{2} \\f^{-1}(f(3))=\frac{\sqrt{\sqrt{4(3)+1} -3}}{\sqrt{2} } \\f(f^{-1} (3))=3[/tex]

What are the inverses of a function?

A function that can change into another function is referred to as being inverse or anti. In other words, the inverse of any function "f" will take y to x if it takes x to y in the first place. The inverse function is indicated by  F^-1 if the function is denoted by F.

Given C={1,2,3}, and

[tex]f(x)=x^{2} +1\\ f^{-1}(x)=\sqrt{x-1} \\f^{-1}(f(x))=\frac{\sqrt{\sqrt{4(x)+1} -3}}{\sqrt{2} } \\f(f^{-1} (x))=x[/tex]

Now,

For C=1,

[tex]f(1)=1+1=2\\f^{-1}(1)=\sqrt{1-1}=0\\f^{-1}(f(1))=\frac{\sqrt{\sqrt{4(1)+1} -3}}{\sqrt{2} } \\f(f^{-1} (1))=1[/tex]

For C=2

[tex]f(2)=4+1=5\\f^{-1}(2)=\sqrt{2-1}=1\\f^{-1}(f(2))=\frac{\sqrt{\sqrt{4(2)+1} -3}}{\sqrt{2} } \\f(f^{-1} (2))=2[/tex]

For C=3

[tex]f(3)=9+1=\sqrt{10} \\f^{-1}(3)=\sqrt{3-1}=\sqrt{2} \\f^{-1}(f(3))=\frac{\sqrt{\sqrt{4(3)+1} -3}}{\sqrt{2} } \\f(f^{-1} (3))=3[/tex]

Hence, the solution is provided.

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Answer:

Step-by-step explanation:

14545

Make a basic proportion

16,000/x=110/100

Cross multiply

1,600,000=110x

Divide

x=14545

Answer:

Step-by-step explanation:

0

Answer:

Step-by-step explanation:

In order for the ratio of volumes of a cylinder, cone, and sphere to be true, the radius of each solid would have to be equal. This is because the radius is a factor in the volume formulas for all three solids:

Volume of cylinder: πr^2h

Volume of cone: (1/3)πr^2h

Volume of sphere: (4/3)πr^3

If the radius is equal in all three solids, then the volumes will be in the same ratio as the given ratio, regardless of the values of h or the surface area. Therefore, the correct answer is : The radius and the height.

Answer:

(2, 5 )

Step-by-step explanation:

5x - y = 5 → (1)

- x + 3y = 13 → (2)

multiplying (1) by 3 and adding to (2) will eliminate y

15x - 3y = 15 → (3)

add (2) and (3) term by term to eliminate y

14x + 0 = 28

14x = 28 ( divide both sides by 14 )

x = 2

substitute x = 2 into either of the 2 equations and solve for y

substituting into (2)

- 2 + 3y = 13 ( add 2 to both sides )

3y = 15 ( divide both sides by 3 )

y = 5

solution is (2, 5 )

Answer:

x = 10

Step-by-step explanation:

To find the value of x:

1) add up like-terms

4 + 2x+1x-4=30

4 + 3x - 4 = 30

2) Add up the number on the left side

0 + 3x = 30

3x = 30

3) isolate the x

3x/3 = 30/3

x = 10

The value of angle a is 45, the value of angle b is 135, the value of angle c is 135 and d is 135.

Straight lines with little depth or width are present. You will learn about a number of lines, including transversal, intersecting, and perpendicular lines. A figure called an angle is one in which two rays originate from the same point. In this area, you might also encounter contrasting and related viewpoints.

Given that the line cutting the two parallel lines. The value of the angles will be calculated as,

Angle a will be,

a + 135 = 180

a = 180 - 135

a = 45

Angle b = 135

Angle c = 135

Angle  d = 135

Therefore, the value of angle a is 45, the value of angle b is 135, the value of angle c is 135 and d is 135.

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Answer:

the fraction 5/7 is equivalent to approximately 71.4%

Step-by-step explanation:

if you divide 5/7, you'll get an answer like 0.71428

to convert decimals to fractions, you have to multiply by 100 to get 71.428

to round to the nearest tenth, round the 2 up to 3

71.43

and then, round 4 down

71.4%

hope this helped!

Answer:

[tex]\huge\boxed{\sf \$ 7.50}[/tex]

Step-by-step explanation:

Given that,

6 inches of sandwich = $2.50

Multiply 3 to both sides

6 × 3 inches of sandwich = $2.50 × 3

[tex]\rule[225]{225}{2}[/tex]

On solving the provided question, we got the area of triangle  - A =  1/2 B X H= 15 sq. units.

A triangle is a shape formed when three straight lines meet. All triangles have three sides and three corners (angles). The point where two sides of a triangle meet is called a vertex. The base of a triangle can be any one of its three sides, but it is usually the bottom one. The longest side of a triangle is opposite its biggest angle, and the shortest side of a triangle is opposite its smallest angle. The three angles inside a triangle always add up to 180°.

To find the Area of triangle -

By help formula,

[tex]A = \frac{1}{2} B X H\\\\[/tex]

[tex]\\A = \frac{1}{2} (6 X 5)\\[/tex]

[tex]\\A = 15 sq.unit[/tex]

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[tex]\textit{equation of a circle}\\\\ (x- h)^2+(y- k)^2= r^2 \hspace{5em}\stackrel{center}{(\underset{2}{h}~~,~~\underset{-3}{k})}\qquad \stackrel{radius}{\underset{4}{r}} \\\\[-0.35em] ~\dotfill\\\\ ( ~~ x - 2 ~~ )^2 ~~ + ~~ ( ~~ y-(-3) ~~ )^2~~ = ~~4^2\implies {\large \begin{array}{llll} (x-2)^2+(y+3)^2=16 \end{array}}[/tex]

The solution to the system of equations is (y,z) = (-23/11, 33/11).

What is the system of equations?

A system of equations is a set of one or more equations involving a number of variables. The solutions to systems of equations are the variable mappings such that all component equations are satisfied in other words, the locations at which all of these equations intersect.

To eliminate the z-terms in this system, you can multiply the first equation by 3 and the second equation by -1 and then add the resulting equations together. This will give you an equation in terms of only y, which you can then solve for y.

The resulting equation will be:

(4y + 3z) * 3 + (y - z) * -1 = (-6) * 3 + (-5) * -1

12y - 3z - y + z = -18 - 5

Combining like terms, we get:

11y = -23

Dividing both sides by 11, we get:

y = -23/11

Now that we have solved for y, we can substitute this value back into one of the original equations to solve for z. For example, substituting -23/11 for y in the first equation gives us:

4(-23/11) + 3z = -6

Solving for z, we get:

z = 33/11

Hence, the solution to the system of equations is (y, z) = (-23/11, 33/11).

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Matching the equation to choices

equation                                                classifications

2x = y + 3 and y = - 2x + 1                       consistent and independent

2x = y + 3 and 2y - 4x = -6                     consistent and dependent

2x = y + 3 and 2y - 4x = 6                       inconsistent

A system is considered to be inconsistent if there is no solution. There is no solution because the line graphs are parallel and do not overlap.

2x - y = 3... equation * 2

4x - 2y = 6

-4x + 2y  = 6

subtracting gives

8x - 4y = 0

no solution  

A system is deemed consistent if it has at least one solution.

A consistent system is independent if it only has one solution.

y = 2x - 3

y = -2x + 1

subtracting gives

0 = 4x - 4

x = 1

substituting x to get y

y = -1

A consistent system is dependent if there are an endless number of possible solutions. The two equations represent the same line when you graph them.

-y + 2x =  3 .... equation * 2

-2y + 4x =  6

2y - 4x = -6  

subtracting gives

-4y + 8x = 12 - none cancels out meaning infinite number of possible solutions

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Answer: 325

Step-by-step explanation:

Lets rewrite this.

35% of sweets are given out of 500.

35% of 500.

35% = 0.35

0.35 x 500 = 175

This means 175 prizes were given. Now we have to subtract.

500 - 175 = 325

The shell from the fireworks will reach the water after 0.75 seconds or -0.625 seconds. Since the time cannot be negative, the shell will reach the water after 0.75 seconds.

To find the time it takes for the shell from the fireworks to reach the water, we need to solve the equation h = 0 for t.

The given equation for h is h = -16t^2 + 64t + 36, so we can substitute this into the equation h = 0 to get:

0 = -16t^2 + 64t + 36

To solve this equation, we can use the quadratic formula, which is:

t = (-b +/- sqrt(b^2 - 4ac)) / (2a)

In this case, a = -16, b = 64, and c = 36, so the formula becomes:

t = (-64 +/- sqrt(64^2 - 4(-16)(36))) / (-32)

Simplifying this gives us:

t = (-64 +/- sqrt(4096 - 2304)) / (-32)

which simplifies to:

t = (-64 +/- sqrt(1792)) / (-32)

Finally, simplifying gives us:

t = (-64 +/- 44) / (-32)

t = (-20 or 24) / (-32)

t = -0.625 or 0.75

Thus, the shell from the fireworks will reach the water after 0.75 seconds or -0.625 seconds. Since the time cannot be negative, the shell will reach the water after 0.75 seconds.

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